Physics Boat Problem involving velocity

[sympathy2]

Homework Statement

1. A river that is 209 m wide flows due east at a uniform speed of 2.2 m/s. A boat with a speed of 8.3 m/s relative to the water leaves the south bank pointed in a direction 30° west of north.
(a) What is the magnitude of the boat's velocity? (m/s)
(b) What is the direction of the boat's velocity relative to the ground? (answer in degrees west of north)
(c) How long does the boat take to cross the river? (s)

The Attempt at a Solution

For part a, I tried a few answers such as 16.6, 7.188, 7.2, 4.15, 7.19, but none of those answers were right. I think my main problem is understanding the question, perhaps i misunderstood the "relative to the water" part. Can someone please explain it to me. The right answer would also be appreciated.

 

[mjsd]

"relative to the water" means in the reference frame of the water. note the water is moving "relative to the ground" (or as seen in the frame of the ground). but if you are IN the ref frame of water, then the "ground is moving" instead. as a result, the velocity relative to water would be different from the velocity relative to the ground, for example.

 

[ogb p]

The boat's velocity can be broken down into two components: one in the east direction, parallel to the river's flow, and one in the north direction, perpendicular to the river's flow. Since the boat's speed is relative to the water, we can use vector addition to find the magnitude of its velocity.

Using the Pythagorean theorem, the magnitude of the boat's velocity can be calculated as:

V = √(8.3^2 + 2.2^2) = 8.6 m/s

For part b, we can use trigonometry to find the direction of the boat's velocity relative to the ground. The angle between the boat's velocity and the north direction can be found as:

θ = tan^-1(2.2/8.3) = 14.8°

Since the boat is pointed 30° west of north, the direction of its velocity relative to the ground is:

14.8° + 30° = 44.8° west of north

For part c, we can use the formula d = v*t, where d is distance, v is velocity, and t is time. We know the distance to be crossed is 209 m, and the boat's velocity is 8.6 m/s. So, the time it takes to cross the river is:

t = d/v = 209/8.6 = 24.3 seconds

Therefore, the boat takes 24.3 seconds to cross the river.